- The pattern as the exponents decrease is that each term is divided by 8 (multiplied by \(1/8\)).
- Applying this pattern:
- \(a = 8^0 = \boxed{1}\)
- \(b = 8^{-1} = \boxed{\frac{1}{8} \text{ or } 0.125}\)
- \(c = 8^{-2} = \boxed{\frac{1}{64} \text{ or } 0.015625}\)
Explanation
- Analyzing the Pattern of Powers of 8
The table displays consecutive integer powers of 8, starting from 8^3 and decreasing the exponent by 1 at each step. By examining the values in the table, we observe:
- \(8^3 = 512\)
- \(8^2 = 64\)
- \(8^1 = 8\)
To find the relationship between consecutive terms as the exponent decreases, we can divide a term by the previous term with the next higher exponent:
- \(64 / 512 = 1/8\)
- \(8 / 64 = 1/8\)
This confirms that as the exponent decreases by 1, the value is multiplied by \(1/8\) (or divided by 8).
- Calculating the Missing Variables
Given the constant factor of \(1/8\), we can determine the values of \(a, b,\) and \(c\) by continuing this pattern from the last known value, \(8^1 = 8\):
- For \(8^0 = a\), we divide the previous term by 8:
\(a = 8 / 8 = 1\)
- For \(8^{-1} = b\), we divide the previous term (a) by 8:
\(b = 1 / 8 = 0.125\)
- For \(8^{-2} = c\), we divide the previous term (b) by 8:
\(c = 0.125 / 8 = 1/64 = 0.015625\)
- Conclusion
In summary, the pattern observed as the exponents decrease is that each consecutive term is found by dividing the previous term by 8. Following this rule, we calculated the values for \(a, b,\) and \(c\) as:
- \(a = 1\)
- \(b = 1/8\) or \(0.125\)
- \(c = 1/64\) or \(0.015625\)
Examples
Understanding powers and exponential growth/decay is essential in many real-world fields, such as finance for calculating compound interest or decay, and in science for modeling radioactive decay or population growth. For instance, if a radioactive substance loses 7/8 of its mass every time period (meaning 1/8 remains), you can use the same logic of dividing by 8 to predict how much of the material will remain after several periods. This fundamental pattern helps in making accurate projections over time.